Puzzle for December 3, 2018 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
From eq.3, substitute D + E + F for B in eq.2: D + E + F + F = C + D + E Subtract both D and E from each side of the above equation: D + E + F + F - D - E = C + D + E - D - E which simplifies to 2×F = C
Hint #2
Replace C with 2×F in eq.4: 2×F - D = D + F Add (D - F) to each side of the equation above: 2×F - D + (D - F) = D + F + (D - F) which means F = 2×D
Hint #3
Replace F with 2×D in eq.4: C - D = D + 2×D Add D to each side: C - D + D = D + 2×D + D which makes C = 4×D
Hint #4
Substitute 4×D for C, and 2×D for F in eq.6: B - D = 4×D + 2×D Add D to each side: B - D + D = 4×D + 2×D + D which makes B = 7×D
Hint #5
Substitute 7×D for B, and 2×D for F in eq.3: D + E + 2×D = 7×D which becomes E + 3×D = 7×D Subtract 3×D from each side: E + 3×D - 3×D = 7×D - 3×D which makes E = 4×D
Hint #6
Substitute 4×D for C and E, and 2×D for F in eq.5: A + D = 4×D + 4×D + 2×D Subtract D from each side: A + D - D = 4×D + 4×D + 2×D - D which makes A = 9×D
Solution
Substitute 9×D for A, 7×D for B, 4×D for C and E, and 2×D for F in eq.1: 9×D + 7×D + 4×D + D + 4×D + 2×D = 27 which simplifies to 27×D = 27 Divide both sides by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 9×D = 9 × 1 = 9 B = 7×D = 7 × 1 = 7 C = E = 4×D = 4 × 1 = 4 F = 2×D = 2 × 1 = 2 and ABCDEF = 974142